MHB Cartesian product and symmetric difference

AI Thread Summary
The discussion focuses on proving the equality Ax(BΔC) = (AxB) Δ (AxC) for three sets A, B, and C. Participants start by defining an arbitrary element p in Ax(BΔC) and explore its components, leading to the conclusion that p belongs to either Ax(B\C) or Ax(C\B). They confirm that this implies p is in (AxB) Δ (AxC), establishing one direction of the proof. The conversation also emphasizes the importance of demonstrating the reverse inclusion by selecting an element from (AxB) Δ (AxC) and showing it belongs to Ax(BΔC). Overall, the discussion successfully navigates the proof of the set equality.
fatineouahbi
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Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .
 
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fatineouahbi said:
Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .

It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?
 
evinda said:
It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?

Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)

Then I'll just try to go backwards maybe ?
 
fatineouahbi said:
Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)
Well done, you are right :)

fatineouahbi said:
Then I'll just try to go backwards maybe ?
Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.
 
evinda said:
Well done, you are right :)

Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.

Thank you so much !
 
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